I Pdc V1.3.0 A Complete Technical Report Including I Pdc, Pmu Simulator, An...
phd thesis
1. POLITECNICO DI TORINO
Corso di Dottorato in Ingegneria Elettronica
e delle Comunicazioni
Tesi di Dottorato
Nonlinear Black-Box Models of
Digital Integrated Circuits via
System Identification
Claudio Siviero
Direttore del corso di dottorato
Prof. I.Montrosset
Tutore: Prof. F.Canavero
XIX Ciclo di Dottorato
2. Summary
This Thesis concerns the development of numerical macromodels of digi-
tal Integrated Circuits input/output buffers. Such models are of paramount
importance for the system-level simulation required for the assessment of Sig-
nal Integrity and Electromagnetic Compatibility effects in high-performance
electronic equipments via system-level simulations.
In order to obtain accurate and efficient macromodels, we concentrate on
the black-box modeling approach, exploiting system identification methods.
The present study contributes to the systematic discussion of the IC mod-
eling process, in order to obtain macromodels that can overcome strengths
and limitations of the methodologies presented so far. The performances of
different parametric representations, as Sigmoidal Basis Functions (SBF) ex-
pansions, Echo State Networks (ESN) and Local Linear State-Space (LLSS)
models are investigated. All representations have proven capabilities for the
modeling of unknown nonlinear dynamic systems and are good candidates too
be used for the modeling problem at hand. For each model representation,
the most suitable estimation algorithm is considered and a systematic analy-
sis is performed to highlight advantages and limitations. For this analysis,
the modeling process is applied to a synthetic nonlinear device representative
of IC ports, and designed to generate stiff responses.
The tests carried out show that LLSS models provide the best overall
performance for the modeling of digital devices, even with strong nonlinear
dynamics. LLSS models can be estimated by means of an efficient algorithm
providing a unique solution. Local stability of models is preconditioned and
verified a posteriori.
The effectiveness of the modeling process based on LLSS representations
is verified by applying the proposed technique to the modeling of real devices
involved in a realistic data communication link (an RF-to-Digital interface
used in mobile phones). The obtained macromodels have been successfully
used to predict both the functional signals and the power supply and ground
fluctuations. Besides, they turn out to be very efficient, providing a signifi-
cant simulation speed-up for the complete data link.
I
3. Acknowledgements
Turin, February 2007
I wish to thank my Ph.D. thesis advisor, Prof. Flavio Canavero for his ad-
vises and comments throughout the whole research activity and the prepara-
tion of this Thesis. I also wish to thanks Prof. Ivano Maio and Dr. Igor Stievano
for supporting my work with continuos helpful suggestions and fruitful dis-
cussions. Their enthusiasm and scientific methodology have been a reference
for me and their contributions improve the quality of the results.
Besides, I wish to thank Prof. Jos´ Carlos Pedro for his excellent guid-
e
ance and Dr. Pedro Miguel Lavrador for his fruitful co-operation during my
foreign experience within the TARGET exchange program at the Instituto
de Telecomunica¸˜es, Aveiro, Portugal.
co
Finally, I would like to thank also Dr. Ali Nadir Arslan for his support
during my research experience at the Nokia Research Center, Helsinki, Fin-
land.
II
7. Chapter 1
Introduction
Present and future high-performance electronic equipments must satisfy higher
and higher design requirements imposed by performance and technology con-
straints. The consequence for the designers is to perform a large number of
Signal Integrity (SI) and ElectroMagnetic Compatibility (EMC) assessments.
Such assessments are of paramount importance in order to detect and cir-
cumvent those sensitive effects like crosstalk, simultaneous switching noise,
immunity and radiation [1] that may seriously compromise the achievement
of the design objectives. This, in turn, implies an increase of the simulations
of the complete system for the prediction of the signals propagation on the
interconnects.
In order to perform such simulations, the combination of propagation ef-
fects with possibly very complex geometry and with the nonlinear behavior
of the active devices makes a direct full-wave approach not feasible. There-
fore, a feasible strategy must subdivide the propagation path into separate
and well-defined sub-systems typically found along the signal propagation
paths, i.e., active devices, transmission-line interconnects, and interconnects
with a complex 3D geometry such as vias and connectors. Each sub-system
is separately characterized by a macromodel, i.e., a set of equations that
are able to reproduce with sufficient accuracy the electrical behavior of the
sub-system. The macromodels are then implemented in a code suitable for
commercial solvers, e.g., SPICE-like, Analog-Mixed-Signal (AMS). Finally,
the simulation of the complete system (i.e., system-level simulation) as a
chain of cascaded blocks is performed within the same environment. The
1
8. 1 – Introduction
obtained voltage or current signals along the propagation paths or the eye-
diagrams and possible other frequency domain figures like the Power Spectral
Density (PSD) are computed and used by the designer to evaluate the system
performance.
It is well known that such analysis is very challenging, since a broadband
characterization of all the sub-systems must be taken into account. Besides,
macromodels have to be efficient in order to require a limited amount of
simulation time and CPU memory. Specifically, the numerical models repre-
senting the active devices play a key role, since they are shaping the signals on
the system interconnections. For the linear interconnects (lumped and dis-
tributed) there are results and well-established methods already published
in literature which lead to accurate and efficient macromodels [2, 3]. On the
contrary, nowadays macromodeling of active devices is a challenging task that
motivates this research activity. The traditional way for devices amounts to
using macromodels based on the internal physical description. Those models
are appropriate to system-level simulation and provide a good result accu-
racy. Unfortunately, the complexity of many devices (e.g., huge number of
variables, unknown nonlinear effects) as well as the lack of information on
their internal structure often makes difficult the development of traditional
physical models. Moreover, whether available, physical models disclose the
Intellectual Property (IP) of the devices and, furthermore, they lead to high
time-consuming simulations demanding a large amount of CPU memory.
In order to overcome the previous limitations, our attention is focused on
the development of Black-Box i.e., behavioral models via system identifi-
cation methods [4]. The behavioral modeling of a system means to look for
a mathematical relation among all the relevant external system variables on
the basis of the external observation of the system response to suitable stim-
uli. No knowledge of the system internal structure is required. Application
examples can be easily found in several other areas of interest. The control
of industrial plants or complex mechanical systems as well as the prediction
of economic phenomena demand the availability of such models. It is worth
to remark that suitable strategies must be devised for modeling devices of
different categories. In this Thesis, we concentrate on the macromodels of
ports of digital Integrated Circuits (ICs), for which the representations based
on nonlinear parametric models are selected. Such an approach has been
2
9. 1 – Introduction
already successfully applied to ICs [2, 5, 6, 7, 8]. Nevertheless, the avail-
able results are rather preliminary and many relevant issues are still open.
Mainly, model stability cannot be easily imposed a-priori or even during
the estimation process without impacting on model accuracy. Furthermore,
higher order dynamical effects may not be readily represented by these mod-
els and model estimation for real devices with multiple ports is troublesome
and affects the quality of the estimated models. As an example, the gen-
eration of device port models including the effects of the neighboring ports
suffers from the increase of complexity of the approximation problem.
Finally, it is worth to mention a complementary activity (not reported in
this Thesis) developed during the triennium of studies. This activity has been
related to the study of macromodels of Radio Frequency (RF) devices [9].
For this research topic, three months were spent at the Wireless Circuits &
Systems Group of Instituto de Telecomunica¸˜es, Aveiro (Portugal) where a
co
methodology for obtaining low-pass equivalent behavioral models of Power
Amplifiers (PAs) has been developed. The results of such an activity are
reported in [10].
The activity of this Thesis mainly contributes to the open issues of the
modeling process for ICs. Specifically
• We investigate the performances of different parametric representation
whose capabilities of modeling unknown nonlinear systems have been
proven by the system identification theory. In particular, such repre-
sentations have been applied for the first time to the IC modeling.
• For each model representation, suitable estimation algorithm are con-
sidered and their advantages and drawbacks are analyzed.
• Finally, the modeling stability issue is addressed.
Outline of the thesis
The structure of this thesis is as follows.
Chapter 2 deals with the general problem of IC ports modeling and
outlines possible modeling approaches.
3
10. 1 – Introduction
Chapter 3 deals with the approach we mainly discuss in this Thesis:
the behavioral modeling via black-box identification. Here we describe the
step-by-step for IC ports by means of parametric models.
Chapter 4 describes and compares possible parametric model representa-
tions suitable for modeling ICs, such as Sigmoidal Basis Functions (SBF) ex-
pansions, Echo State Networks (ESN) and Local Linear State-Space (LLSS)
models.
Chapter 5 reports the systematic study of the performances of the pro-
posed parametric representations to the modeling of a synthetic nonlinear
dynamic one-port test device. From this test, the parametric representation
providing the best overall performance is selected.
Chapter 6 discusses the impact of the proposed macromodel represen-
tation to the system-level simulation of data-link for mobile applications.
Finally, Chapter 7 provides the main conclusions that can be drawn
from the results of this Thesis.
4
11. Chapter 2
IC macromodels
2.1 Introduction
This Chapter outlines possible modeling approaches for the development of
macromodels of ICs suitable for system-level simulation. As an example,
Figure 2.1 shows the typical block diagram of a digital IC, mainly composed
of the internal logic core and the input/output buffers driving and loading
the interconnects themselves. Since digital ICs are complex systems, contain-
ing a very complex logic core and a high number of pins (several hundreds
for modern microprocessors), there is no hope to effectively model both the
IC internal logic and the input/output ports. For this reason and for the
prediction of waveforms on interconnects, we require effective and accurate
macromodels of digital IC ports. Since digital input/output buffers are non-
linear dynamic devices, we are interested in the development of macromodels
described in the time-domain.
Ú
¾ ÐÓ
Ò¾
Ú¾
ÓÖ
½ Ó½
Ò½ ÓÙؽ
Ú½ ÚÓ½
Figure 2.1. Block diagram of a generic digital IC.
5
12. 2 – IC macromodels
In order to be effective for system-level simulation, IC macromodels must
fulfill the following requirements
(i) Intellectual Property (IP) protection: macromodels should not disclose
the information on the internal structure and technology of devices, in
order to discourage any attempts of reverse engineering.
(ii) Accuracy: macromodels must provide responses in agreement with re-
spect to the reference device , in particular for the prediction of higher
order effects.
(iii) Efficiency: macromodels involved in system-level simulation have to
consume as less simulation time as possible, as well as a limited amount
of CPU memory.
(iv) Implementation in any commercial simulator: macromodels must be
easily translated in code for each platform being used.
It is worth to remark that the development of macromodels meeting all
the above mentioned requirements is a very challenging task. Basically,
macromodels can be classified in two classes: physical models and behav-
ioral models. Specifically, a physical model is based on the reconstruction of
the internal structure of the device, whereas behavioral model is defined as
a set of port characteristic equations (or the equivalent circuit of such equa-
tions) obtained from external (possibly virtual) measurements. In the next
Sections we discuss the basic approaches belonging to the two methodologies,
by highlighting their strengths and limitations.
2.2 Transistor-level models
The traditional way for IC ports modeling amounts to describe the device
behavior by means of a detailed physical model based on internal structure.
This is the so-called transistor-level description. As an example, Figure 2.2
shows the circuit of a typical 4-stages CMOS 1.2 µm output buffer, whose
output terminals are (a) and (b) [11]. In this circuit, vi denotes the buffer
input voltage (i.e., the output of the logic core of the IC), v and i are the
buffer voltage and current at the output pin, respectively, and Vcc and Vss
indicate the power supply voltages.
6
13. 2 – IC macromodels
Î
ÄÔ ´ µ
Ú Ô Ú
Î×× ´ µ
Figure 2.2. Example circuit for a typical 4-stages 1.2 µm CMOS output
buffer [11].
This transistor-level model is the most accurate solution, and usually
it is considered as the reference for the device. Unfortunately, the model
completely discloses the internal details of the device, unless the developer
releases an encrypted version of the transistor-level. In addition transistor-
level descriptions are generally large in size and their effect is to slow down
the simulation. Finally, they can not be easily plugged in any simulator,
since they are usually written for a specific simulator in particular when the
encryption holds.
2.3 Simplified equivalent circuits
In order to overcome the transistor-level limitations, behavioral modeling
appears to be effective. The most common approach to behavioral modeling
is via simplified equivalent circuits. A port model is defined by a suitable
equivalent circuit, whose parameters can be estimated from input/output
data. The information on the IC technology is used to devise the equivalent
circuit, i.e., the model structure. An important example of the equivalent
circuit approach to behavioral modeling is the widely adopted Input/output
Buffer Information Specification IBIS). IBIS is a set of rules defining and
formatting data, from which IC port models based on simplified equivalent
circuits can be developed [12]. IBIS offers high numerical efficiency, large data
library and commercial software tools handling models and complex modeling
problems. However, the equivalent circuit approach to behavioral modeling
has also inherent limitations. Mainly the estimation of model parameters is
easy only by virtual measurements, i.e., from transistor-level models of the
7
14. 2 – IC macromodels
devices, and the physical effects taken into account by the model are decided
a priori, when the equivalent circuit defining the model is selected.
Vcc
fH (v) Rpkg Lpkg i
fL (v) Ccomp Cpkg v
Figure 2.3. IC output port equivalent circuit assumed by IBIS.
As an example, Figure 2.3 shows the simplified equivalent circuit as-
sumed by IBIS for the output port of a generic digital IC like the one shown
in Figure 2.2. In this simplified circuit, the electric equivalent of the pack-
age is composed of the Rpkg , Lpkg , Cpkg elements and the silicon output port
capacitance is assumed linear and modeled by Ccomp . Finally, the voltage-
controlled current source fH and fL account for both static characteristics
when the port is driven either in LOW or HIGH output state and the dy-
namics during state switching. Data provided by IBIS, in accordance to
the assumed equivalent circuit, must be translated into an executable model
(IBIS model) in order to be used in circuit simulation environments.
2.4 Parametric models
A second possible approach to behavioral modeling, the one we mainly ad-
dress in this Thesis, is via parametric models and system identification meth-
ods [4]. Such an approach amounts to the selection and estimation of a
suitable nonlinear dynamic parametric model from the waveforms that can
be measured at the IC ports. As an example, a general parametric model
description of the IC output port in Figure 2.2 writes
i(t) = F (Θ , v(t) , d/dt) (2.1)
8
15. 2 – IC macromodels
where F is a suitable mathematical representation depending on the para-
meters collected in vector Θ.
In this approach, the modeled device is considered as a Black-Box, i.e.,
in principle, no knowledge of the internal structure is required and the mod-
eling information is completely contained in the device external responses.
Owing to this feature, parametric models do not disclose any IP information
and can be effectively estimated from measured transient responses or from
simulated responses computed for detailed reference transistor-level models.
Parametric models offer high accuracy and an acceptable numerical efficiency.
Furthermore, since the model structure is selected by the estimation process
itself, parametric models automatically take into account all the physical ef-
fects relating input and output data. In addition, they can be effectively
translated in any commercial simulator. For such motivations, parametric
models would enable any user to easily model sample devices and to simulate
critical interconnect structure to assess sensitive SI/EMC effects.
2.5 Summary
Parametric modeling by means of Black-Box identification techniques ap-
pears to be appealing for the modeling of IC ports. This is also motivated
by the system identification theory, that provides methodologies for develop-
ing effective parametric models of any unknown nonlinear systems. In fact,
several results have been published on the identification of systems in the
control automatic and mechanical area [4, 13]. Indeed, parametric modeling
has been recently applied to ICs with good results [2, 5, 6, 7, 8]. Nevertheless,
the modeling process still presents many open research issues. Some of them
are addressed in the next Chapters.
9
16. Chapter 3
Parametric modeling
3.1 Introduction
In this Chapter we describe the step-by-step procedure for developing para-
metric model of ICs. For the sake of simplicity, in the following general
description we focus on an IC port with external voltage and current v and i
respectively, as shown in Figure 3.1 (the extension to multiport case is trivial
and does not modify the results presented in the next sections).
i(t)
IC v(t)
Figure 3.1. Example of IC port under modeling
The parametric modeling procedure of an IC port can be divided into five
steps described in the following sections.
(1) Model selection: the starting point amounts to selecting the functional
form of model equation, referred to a model representation.
(2) Identification signals: the port needs to be driven by specific signals
in order to obtain transient voltage and current signals carrying infor-
mation on its behavior. The excitation (input) and response (output)
signals involved in this step are named identification signals. It is worth
10
17. 3 – Parametric modeling
to notice that the identification setup can be reproduced by using either
the real device or its detailed transistor-level model.
(3) Model estimation: this step amounts to the numeric computation of the
model parameters so that the model responses mimic well the identifi-
cation signals.
(4) Model validation: once the model has been estimated, an assessment
of the accuracy of the model to reproduce the responses to different
excitations, is needed.
(5) Model implementation: finally, the translation of the obtained model
in a standard circuit simulation environment is performed.
3.2 Model selection
The selection of the parametric model representation is the crucial point of
the modeling process, since good models arise only when the model represen-
tation is suitable for the system being modeled. The model representation
suitable for an IC port is searched within the class of discrete-time para-
metric models. This is mainly due to the large availability of resources for
the estimation of this class of models [4]. Besides, this is the natural choice
when the raw data, i.e., the external responses of the IC port, are known as
sampled waveforms. A very general and compact equation of a discrete time
parametric model for the IC port in Figure 3.1 is
i(k) = F (Θ ; v(k)) (3.1)
where k is the discrete time variable, F is the nonlinear mapping describing
the model and Θ is the vector collecting the parameters. Equation (3.1)
describes a unified framework to handle a large number of discrete-time
parametric representations. In particular, system identification literature
provides a large number of available representations for F that could be
applied to the modeling of unknown nonlinear dynamic systems. Typical
examples are neural networks based on radial, sigmoidal or spline basis func-
tions, wavelet decomposition, kernel estimators, fuzzy models [13], support
11
18. 3 – Parametric modeling
vector machines [14], composite local-linear models [15], Wiener or Volterra
polynomials [16], Wiener-Hammerstein models [17].
It is worth to remark that the model for IC ports can be also represented
as a sum of a nonlinear static part and a nonlinear dynamic part [18], e.g.,
i(k) = Fs (v(k)) + Fd (Θ ; v(k)) (3.2)
where Fs and Fd are the possible static and dynamic parts of the model,
respectively. The splitting of static and dynamic contributions is justified
in Appendix A. The static part can be easily obtained from measurements,
whereas parametric representation of the form (3.1) can be used for Fd . In the
following, denoting splitted structures or models, we refer to equation (3.2),
and using fully nonlinear terminology we refer to equation (3.1). It is of
worth to note that the splitted representation turns out to have practical ad-
vantages in some applications, since it facilities the estimation of Fd [19] and
may contribute to better models for some critical ports exhibiting strongly
nonlinear dynamics.
3.3 Identification signals
Once the model representation F is chosen, the next step of the modeling
process amounts to driving the IC port to obtain transient voltage and cur-
rent signals carrying information on its behavior. The excitation (input) and
response (output) signals involved in this step are named identification sig-
nals. As an example, Figure 3.2 shows the typical setup for collecting the
identification signals for an IC port. The voltage source vs (t) is connected
to the port via a resistor Rs in order to obtain a non-stiff transient test. In
this case the recorded port voltage v (t) corresponds to the excitation and the
¯
recorded port current ¯(t) corresponds to the response.
ı
As a general rule, the driving waveforms (input identification signals)
must be carefully designed in order to excite every possible dynamic behav-
ior of the port [4]. For linear systems, this is easily accomplished by using
input identification signals with a frequency content that spans the frequency
interval containing the system poles; generally white noise or pseudo-random
binary signals are used. For the nonlinear case, unfortunately, only qualita-
tive guidelines are available for the design of the input identification signals.
12
19. 3 – Parametric modeling
¯(t)
ı Rs
IC v (t)
¯ vs (t)
Figure 3.2. Typical identification setup for the IC port of Figure 3.1
Such signals should contain large steps with rise times short enough to ex-
cite the fast dynamic behaviors of the system and flat levels allowing the
system to approach steady state operations on several operating points. A
superimposed small noise signal usually improves the ability of such signals
to excite the system dynamics. The final results are multilevel signals with
superimposed small noise as the one reported in Figure 3.3.
vs(t)
VDD
0 t
Figure 3.3. Example of a typical multilevel signals with superimposed
noise used for vs in Figure 3.2. In this case the voltage level VDD is referred
to the value of the power supply voltage for the IC under modeling.
Of course the rise times of the steps and the durations of the flat parts
must be tuned on the fastest and slowest time constants that can be observed
in the system responses, and, as a further rule of thumb, the number of
different levels should increase, as the nonlinearity of the static characteristic
becomes stronger. Again, the design of the input identification signals is a
matter of repeated estimation experiments. Specific guidelines required for
the generation of identification signals for the modeling of IC buffers are
provided in [5]. In addition a systematic study of the effects of estimation
13
20. 3 – Parametric modeling
signals on the quality of IC buffer models is reported in [19].
3.4 Parameters estimation
The parameter estimation is obtained by means of standard methods, and
amounts to fitting the response of the model to the identification signals
recorded in the previous step. The simplest fitting approach is to look for Θ
minimizing the mean square error between the model and the port responses.
This means to find
N
1
Θ | min ( ¯(k) − i(k) )2
ı (3.3)
N k=1
where ¯(k) = ¯(kTs ) is the sampled output identification signal, Ts is the
ı ı
sampling period, N is the total number of samples and i(k) is the response
of model (3.1) to the sampled input identification signal v(k). The sampled
¯
identification signals must contain all the information of the original iden-
tification signals. Therefore the sampling period Ts must be smaller than
the sampling time TN defined by the Nyquist frequency of the identification
signals. On the other hand, the sampling time should not be too small, in
order to avoid oversampling and consequent numerical problems in the min-
imization of (3.3). As a rule of thumb, the ratio TN /Ts should be on the
order of 2 ÷ 6.
The minimization of (3.3) amounts to solving a nonlinear least squares
problem. For this task, iterative algorithms are proven to be particularly
effective. Nonlinear optimization literature provides specific algorithms de-
pending on the choice of the representation F . Typical resources are gradient-
based methods [20], genetic algorithms [21], extended Kalman algorithm [22],
simulated annealing [23]. It is worth to remark that the problem (3.3) is
nonconvex and the solution obtained by means of the iterative algorithm is a
local minima. Thereby, a good initial estimate of the model is of paramount
importance, since the initial starting point determines in which local minima
the algorithm will end up.
14
21. 3 – Parametric modeling
3.5 Model validation
The estimation procedure previously described yields, along with the best
values of the model parameters, the error of the model in reproducing the
output identification signal. Such an error is the first indication of the model
quality, because large errors at this stage imply the failure of the modeling
process. However, the performances of such models must be further assessed
by checking their response to an input signal different to the one used for the
identification. In fact, models that reproduce well the identification signal
may still exhibit strange behaviors when the input signal is changed. Specif-
ically, model validation addresses the following issues: (i) accuracy i.e., the
error between the validation reference response and model response; (ii) effi-
ciency i.e., speed-up of the computation of the model response; (iii) stability
i.e., model response should not account for spurious dynamic behaviors (e.g.,
oscillation, saturation) not present in the modeled device.
It is worth to remark that unstable models must be avoided, even if they
reproduce the reference responses well. In fact, numerical simulation of these
models for different signal and load conditions may lead to possible unstable
behavior. When dealing with nonlinear model, stability analysis is not a triv-
ial issue [24]. In particular the formulation of a general stability requirement
is a challenging task. The application of a global stability criterion seems to
be too complicate and likely too restrictive. On the other hand, a stability
study based on a limited class of excitations and load conditions would be
suitable. Nevertheless, this approach would not provide an exhaustive valida-
tion of the model. In order to devise a useful and feasible stability criterion,
we study the local stability of the parametric model, as suggested in [25].
This methodology seems to be the simplest and it is readily extended from
the linear case. Local stability analysis relies on the linearization of the model
over each time step of the transient validation test. Each linearized model is
then represented in the equivalent state-space form and the eigenvalues of the
matrix mapping the states are computed (see more details in Appendix B).
The parametric model is considered locally stable if the computed eigenval-
ues lie within the unitary circle for each time step. However, several test
carried out have shown that the presence of some instants of time in which
the eigenvalues lie outside the unitary circle does not significantly affect the
15
22. 3 – Parametric modeling
final performance of the model. The percentage of points of the validation
set having at least one eigenvalue outside the unitary circle over the total
number of points can be used as a confidence measure for the stability of the
model. This ratio can be used as a criterion to assess the local stability of
the obtained model. The smaller the ratio the smaller the probability that
the model will become unstable during its operational evolution. Therefore
model generation should aim obtaining models not showing any eigenvalue
outside the unitary circle.
3.6 Macromodel implementation
The last part of the modeling process is the synthesis of the estimated
discrete-time parametric model defined by (2.1) as an equivalent circuit to
be implemented in standard simulation environments. For the sake of con-
ciseness, and without loss of generality, we concentrate on the description of
the following synthetic nonlinear discrete time parametric model
i(k) = a1 i(k − 1) + b0 e−c0 v(k) + b1 v(k − 1) (3.4)
where a1 , b0 , c0 and b1 are the parameters. For the implementation of the
above model as macromodel, it is useful to convert equation (3.4) into a
continuous-time state-space realization [5], as described in the following.
Equation (3.4) is rewritten into the discrete-time state-space form
x1 (k) − x1 (k − 1) = a1 x1 (k − 1) + b0 e−c0 v(k−1) + b1 x2 (k − 1)+
−x1 (k − 1)
(3.5)
x2 (k) − x2 (k − 1) = v(k) − x2 (k − 1)
i(k) = a1 x1 (k) + b0 e−c0 v(k) + b1 x2 (k)
where x1 (k) = i(k − 1) and x2 (k) = v(k − 1). The difference operator
in (3.5) is then approximated with a differential one, (e.g., (d/dt)z(t) ∼ =
(1/Ts ) (z(k) − z(k − 1))). In this way, the time variable t is restored and the
final equivalent continuous-time state-space representation arises
d x1 (t) 1 a1 x1 (t) + b0 e−c0 v(t) + b1 x2 (t) − x1 (t)
=
dt x (t)
Ts v(t) − x2 (t)
2
(3.6)
i(t) = a x (t) + b e−c0 v(t) + b x (t)
1 1 0 1 2
16
23. 3 – Parametric modeling
In order to use models expressed by (3.6) for the numerical system-level sim-
ulations, two practical choices are available: 1) convert the equations into
circuit equivalents and exploit a SPICE circuit simulator [26]; 2) directly
implement their expressions in analog mixed-signal (AMS) simulation envi-
ronments, like Verilog-AMS [27] and VHDL-AMS [28, 29], that accept and
solve differential-algebraic equations.
The conversion of differential-algebraic equations into circuit equivalents
and their implementation as SPICE subcircuits is a standard procedure [5].
To do this, the first two rows of (3.6) can be implemented by simple equiv-
alent circuits with voltage controlled sources and the third one by a current
controlled source only. As an example, Figure 3.4 shows the circuit synthesis
of the second equation of (3.6). The circuit synthesis of the first equation is
obtained by properly replacing the controlled source. The complete equiva-
lent circuit of (3.6) can be easily coded as a SPICE-like subcircuit, as shown
in Figure 3.5.
R
v(t) C x2 (t)
d 1
Figure 3.4. RC equivalent circuit for dt x2 (t) = Ts [v(t) − x2 (t)], Ts = RC.
3.7 Open issues
Nowadays, the parametric modeling process described in this Chapter presents
several issues that have not been fully investigated yet. In particular, re-
ferring to the five steps outlined in the Introduction of this Chapter, the
following issues need consideration.
(1) Model selection: properties, advantages/disadvantages and applicabil-
ity range of a comparative analysis of different parametric represen-
tation need to be investigated; in particular, their suitability for IC
modeling has still to be determined.
17
24. 3 – Parametric modeling
.subckt macromodel v ref
+ PARAMS:
* sampling time Ts=Rx*C (Rx=1, C=Ts)
+ Rx = 1
+ Ts = ...
* model parameters
+ a1 = ... b0 = ... b1 = ... c0 = ...
* dx1/dt={y-x1/Ts}
Cx1 x1 0 {Ts}
R1 x1 z1 {Rx}
Ex1 z1 0 value={V(y)}
* dx2/dt={v-x2/Ts}
Cx2 x2 0 {Ts}
R2 x2 z2 {Rx}
Ex2 z2 0 value={V(v,ref)}
* output controlled current source i(t)
Gy v ref value={V(y)}
* model representation/structure
EF y 0 value={a1*V(x1)+b0*EXP(-1*c0*V(v,ref))+b1*V(x2)}
RF y 0 {Rx}
.ends
Figure 3.5. SPICE implementation of the parametric model (3.4), via the
representation (3.6).
(3) Model estimation: for each representation, estimation algorithms and
the assessment of their performances require further investigations.
(4) Model validation: a thorough stability analysis of models is required.
No extra efforts are planned for step (2) because, so far, multilevel signals
seem to be appropriate for the IC modeling. However, a study of possi-
ble improvements provided by excitations of different shape can surely be
an interesting topic for future work. Finally, also step (5) does not need
18
25. 3 – Parametric modeling
further consideration since the macromodel implementation is essentially a
well-established technicality.
The aim of the following Chapters is the development of studies and
contributions related the outlined open issues.
19
26. Chapter 4
Model representations
4.1 Introduction
In this Section we investigate possible parametric model representations for
the modeling of ICs by means of the process described in Chapter 3. For
the sake of simplicity, our discussion deals with the approximation of the
constitutive relations of the IC port described by an external voltage v and a
current i, as shown in Figure 3.1. Even if unknown, the constitutive relation
of the IC port can be described by an arbitrary continuous-time state-space
representation involving external measurable variables and internal nonmea-
surable state variables, as follows
˙
x(t) = g (x(t) , v(t))
(4.1)
i(t) = f (x(t) , v(t))
where x is the vector of internal state variables, and g and f are multivariate
nonlinear mappings.
For the approximation of the input-output behavior of (4.1), the identifi-
cation literature provides many parametric model representations [13, 14, 16,
17, 30]. Such models show the so-called universal approximation capability,
i.e., for a sufficiently large size of the model an arbitrarily small modeling
error can be achieved. According to Section 3.2, the typical parametric to
be considered are discrete-time relations. Specifically, such models can be
represented by a Nonlinear Input-Output (NIO) description or alternatively
by a Nonlinear State-Space (NSS) description. Both classes have strengths
20
27. 4 – Model representations
and limitations, as well as the methods available in literature for the estima-
tion of their parameters. For the modeling problem at hand, it is not clear
what the real benefits of using NIO rather than NSS models are. Therefore,
a systematic study of the two classes is needed.
In this Chapter, the two representations are introduced and discussed in
details. In particular, the next sections present the mathematical structure of
the models, their general strengths (well known from the literature and from
the application of these models to real problems) and the available methods
for parameter estimation. All the models considered are good candidates for
the IC modeling. It is worth to remark that in the following section we focus
on widespread model representations available in literature, that have been
proven to provide good results in applications, other than the one addressed
in this Thesis. A systematic comparative study for the selection of the best
model representation is deferred to Chapters 5 and 6.
4.2 Classification: NIO and NSS models
In this section we present and discuss the features of the NIO and NSS
parametric model classes.
Most single-input single-output NIO parametric models can be written
as [13]
i(k) = F ( Θ ; ϕ(k) )
(4.2)
ϕ(k) = [ i(k − 1) , . . . , i(k − r) , v(k) , . . . , v(k − r) ]T
where k refers to the discrete-time variable, i(k) is the output sequence of
the model, v(k) is the input sequence, F is a nonlinear mapping defining the
model representation, and Θ is the vector of model parameters. Vector ϕ(k)
is named the vector of regressors collecting the past r samples of the output
and the present and past r samples of the input, r being the dynamic order of
the model. As outlined in [13], equation (4.2) provides a unified framework
to handle models from both system identification area and from other areas
like neural networks, wavelets and fuzzy systems.
On the other hand, a generic single-input single-output NSS parametric
21
28. 4 – Model representations
model writes
z(k) = F1 ( Θ1 ; z(k − 1) , v(k − 1) )
(4.3)
i(k) = F2 ( Θ2 ; z(k) , v(k) )
where i(k) and v(k) are the output and input sequence, and
z(k) = [z1 (k) , . . . , zn (k)]T (4.4)
is the virtual state vector. The state vector (4.4) is not necessarily an estimate
of the real internal state x of (4.1), even if ideally it could be but, a priori,
z and x are not even correlated. Model (4.3) is defined by the multivariate
nonlinear mappings F1 and F2 depending on the vectors of parameter Θ1
and Θ2 respectively.
Although the model representations (4.2) and (4.3) can be related to each
other [30], the two classes exhibit different features.
NIO models have been successfully applied to real modeling of nonlin-
ear systems with nonmeasurable states in the area of automatic controls.
They involve only the input and output measurable variables and reduce
the dependence on the nonmeasurable states to a direct dependency on the
output dynamics. Furthermore, the estimation of such a class of models can
be done by means of well-established methods. All these motivations have
driven the interest in NIO models as good candidates for IC modeling. NIO
models have been recently proven to accurately reproduce the behavior of
a wide class of commercial devices [5, 6, 7]. In particular, they turn out to
be very compact, thus leading to models with a very small size. Owing to
this, the estimated models, implemented in a simulation environment, are
very efficient and allow simulation speed-ups on the order of 10 ÷ 1000 with
respect to the transistor-level descriptions of devices. Nonetheless, NIO re-
lations have shown inherent limitations. Mainly: (i) model estimation for
real devices with multiple ports is troublesome and impacts on the quality of
estimated models; (ii) higher order dynamical effects may not be readily rep-
resented by these models; (iii) model accuracy depends on the initial guess
of parameters and on local minima of the cost function; (iv) local stability
of models can not be easily imposed a-priori or even during the training
process without impacting on model accuracy. It is worth to remark that,
according to Section 3.5, locally unstable models must be avoided, even if
22
29. 4 – Model representations
they well reproduce the reference responses used in the model estimation. In
order to address the previous limitations, NSS models appear to be a use-
ful alternative. A state-space model is the most natural representation of a
dynamic system. Besides, NSS models have been widely used in the area
of automatic controls and system theory to model actual nonlinear systems
such as industrial plants or power electric machines. The specificity of NSS
modeling arises from the fact that the internal state variables gives more
flexibility with respect to NIO models. However, NSS models have a compli-
cate structure due to the presence of a larger number of degrees of freedom.
Moreover, the related estimation methods are still under study.
As outlined in Section 4.1, in this activity we concentrate on different
example representations belonging to both NIO and NSS classes. All the
considered representations have potential strengths and need to be further
investigated for our problem. In particular, the representations considered
in the systematic study carried out in this thesis are the following
(i) Representations of the NIO class. A general way to define nonlinear
mappings F of NIO models (4.2) is to exploit sums of nonlinear func-
tions of regressors [13]. Many different basis functions can be used,
giving rise to model representations with significantly different proper-
ties. In the PhD thesis [8], macromodels based on Gaussian Radial Ba-
sis Function (RBF) expansions [13] have been studied and successfully
applied to the macromodeling of the ports of digital ICs [5, 6]. Such
models offer remarkable advantages. Mainly, they are robust and have
a regular and smooth behavior outside the fitting domain and the es-
timation of model parameters relies on simple and efficient algorithms.
However, for the problem at hand, macromodels based on Sigmoidal
Basis Functions (SBF) expansion turned out to be more effective [7].
SBF models show properties that are more suitable for fitting the ac-
tual constitutive relations of IC ports and usually lead to simpler (more
efficient) macromodels than those based on RBF. Furthermore, the al-
gorithms themselves for the estimation of SBF models, even if require
more complex and fully nonlinear procedures, allow for more accurate
estimates. This points to exploiting only the SBF models within the
NIO class; Section 4.3 provides further details on SBF models.
23
30. 4 – Model representations
(ii) Representations of the NSS class. Discrete-time NSS models have been
recently studied in literature for the identification of nonlinear dynamic
systems and only preliminary results are available [17, 25, 30, 31, 32,
33, 34, 35]. Within the available options, in this study we concentrate
on two representations that seem to be attractive: the Echo State Net-
works (ESN) [34, 36] and the Local-Linear State-Space (LLSS) mod-
els [35, 37].
4.3 Sigmoidal Basis Functions (SBF)
Models based on SBF describe F in equation (4.2) as an expansion of sig-
moidal functions. The general input-output relation of a SBF model writes
p
i(k) = αj Φ aT ϕ(k) + bj
j (4.5)
j=1
where Φ is the sigmoidal mother function. In this study Φ = tanh. In
equation (4.5), p is the number of basis functions (size of the model) αj
are linear coefficients, and aj and bj are the nonlinear parameters of the
sigmoidal function. In a different and equivalent perspective, the above model
is a recurrent neural network with one hidden layer and a linear output
unit [13, 38, 39].
The estimation of the parameters of model (4.5) requires the solution of
the nonlinear problem defined by (3.3). Within the many possible estimation
algorithms we found good results with the well-known Levenberg-Marquardt
(LM) based methods [40] in conjunction with the pseudo-random procedure
for the selection of the initial guess of parameters, as suggested in [41]. Specif-
ically, we consider two different implementations of the above algorithm, re-
ferring to the two schemes most diffused in literature for the estimation of
NIO models. The first version is called Static LM (SLM) [42]. This procedure
refers to the most common scheme for the estimation of static feedforward
neural networks [38]. In fact, this method solves a nonlinear static problem
i.e., the past samples of the model output accounted in ϕ(k) are forced to
be the samples of the output identification sequence. On the contrary, the
second version called Recurrent LM (RLM) [43] solves the problem consider-
ing the recurrent nature of the model i.e., ϕ(k) accounts for the feedback of
24
31. 4 – Model representations
the model output. The mathematical details of both versions applied to NIO
structures are reported in Appendix C. It is worth to notice that neither
SLM nor RLM version of the algorithm address the model stability issue.
Therefore, SBF model stability can be only verified a-posteriori without any
guarantee of obtaining a local stable model.
The starting point of the estimation procedure for SBF models is to define
completely the model representation by setting the size parameters i.e., the
dynamic order r and the number of basis functions p. The dynamic order
is rather a property of the device under modeling and can be determined
a priori from the device responses [44] or simply postulated and verified a
posteriori. In order to decide the most suitable size p, each estimation run
consists of a comparison of models with increasing p value (e.g., 1 ÷ 10), with
respect to the estimation accuracy. As a rule of thumb, good p values are
the smallest ones leading to a good reproduction of the identification signals.
Suitable statistical indexes help the selection of p [4].
4.4 Echo State Networks (ESN)
This Section briefly discusses the main features of ESN modeling. This class
of models has been recently presented in the literature and provides very
good results for the modeling of the complex dynamic behavior of real sys-
tems [34, 36, 45]. A comprehensive discussion of ESN can be found in [34]
and references therein. This model belongs to the discrete-time NSS class.
As an example, for the system in (4.1), an ESN parametric representation
writes
z(k) = F1 (Az(k − 1) + bv(k − 1))
(4.6)
i(k) = cT z(k) + c0 v(k)
where vector z = [z1 ,...,zn ]T collects the n internal state variables. Typical
values of n are in the range [20, 500] and the nonlinear multivariate mapping
F1 consists of the collection of sigmoidal functions. In this study, F1 =
[ tanh , . . . , tanh ]T . The parameters of the model are the square matrix
A, vectors b and c, and the scalar c0 .
It is worth noting that the estimation of (4.6) would in principle require
the solution of a nonlinear optimization problem. This limitation, along with
the large number of unknown parameters involved in a state-space equation,
25
32. 4 – Model representations
would limit this approach for the identification of complex nonlinear dynam-
ical systems. In order to address the previous limitations, the methodology
proposed in [34] amounts to adapt only the weights of the network-to-output
connections, i.e., the vector c and the scalar c0 , yet leading to the solution of
a linear least squares problem. The other parameters, i.e., matrix A and vec-
tor b, are defined a-priori in a way that allows the inclusion in the model of
a large number of randomly generated dynamics, possibly including those of
the original system under modeling. Matrix A is chosen to satisfy the “echo
state” property, which means that for each input sequence, model (4.6) must
present a unique sequence of state variables. In order to fulfill the “echo
state” requirement, the square matrix A is chosen to have a spectral ra-
dius ρ (i.e., the largest magnitude of the eigenvalues) smaller than one (e.g.,
ρ = 0.7 ÷ 0.99). It can be easily proven that the above assumption, along
with the choice of the tanh function in the nonlinear mapping F1 , leads to lo-
cally stable ESN models. Within the many possible choices for A, that have
not been completely investigated yet, a possible solution is proposed in [34].
In the above paper, matrix A is generated as a sparsely and randomly con-
nected matrix, whose elements are independent random variables that have
a certain (high) probability to be zero (e.g., 95%), and the complementary
probability, labeled as connectivity (e.g., 5%), to be ±a, where a is real pos-
itive number. The number a is then computed to enforce the spectral radius
ρ to the desired value. This choice satisfies a sufficient and hence restrictive
condition for the “echo state” property. Better solutions allowing a higher
number of degrees of freedom exist and are currently under investigation [46].
The details on the criteria used for the initialization of the model and for the
estimation algorithm are reported in [34]. Basically, the only size parameter
to be tuned during the estimation, is the size n of the state vector. Roughly
speaking, a suitable value of n can be selected by performing several model
estimation runs with increasing n (e.g., n = 20 , 50 , 100 , . . .) and then
selecting the lowest value of n providing a sufficient estimation accuracy.
4.5 Local Linear State-Space (LLSS)
The idea underlying the LLSS modeling methodology is the approximation
of the complex dynamic behavior of a nonlinear dynamic system by means
26
33. 4 – Model representations
of the composition of local linear models [15]. The whole operating range
of the system is partitioned into smaller operating regions where the system
behavior is approximated by a linear state-space equation. Even if this idea
has been already investigated in the literature, the implementation presented
in [35, 37] has several strengths, including the nice feature of providing the
automatic computation of local linear models as well as the generation of
the weights for the local models from input-output system responses only.
Furthermore, the special structure of LLSS models facilitates the estimation
procedure. As an example, for the system (4.1) a LLSS model is defined by
the following discrete-time NSS representation
p
z(k) =
ρj (s(k − 1)) ( Aj z(k − 1) + bj v(k − 1) + oj )
j=1
p (4.7)
T
i(k) =
ρj (s(k)) cj z(k) + dj v(k) + qj
j=1
where p is the number of local-linear models, vector z = [z1 , . . . ,zn ]T collects
the n internal states and ρj (·) is the weighting coefficient of the j-th local
model. Each local model is defined by the state matrix Aj and by vectors
bj , oj and cj and by scalars dj and qj . The argument of the weights, i.e.,
the scheduling vector s(k), corresponds to the operating point of the system
and is in general a function of both input and state variables. Among the
possible choices for s(k), a common solution in local linear modeling (also
used in [35, 37]) amounts to collecting the present and past samples of the
input sequence v(k) only. This writes
s(k) = [ v(k) , v(k − 1) , . . . , v(k − r) ]T (4.8)
where r refers to the chosen number of past input variable samples.
It is common practice in local linear modeling to use normalized radial
basis functions for the weights ρj (s(k)) i.e., each weight varies between zero
and one and their sum is forced to be one at each operation point of the
system. Therefore the j-th weight writes
φj (s(k))
ρj (s(k)) = p (4.9)
φi (s(k))
i=1
27
34. 4 – Model representations
where φj (·) is the j-th radial basis function defined as
2
s(k) − tj
φj (s(k)) = exp − 2
(4.10)
βj
Each radial basis function (4.10) is defined by its position in the space of the
scheduling vector (center tj ) and by its spreading (scale parameter βj ).
For the computation of model parameters, i.e., the local model parame-
ters in (4.7) and the parameters defining the weights in (4.10), we perform the
solution of the nonlinear optimization problem by applying the LM method
for NSS structures (see details in Appendix C). In particular, the LM ver-
sion proposed in [35, 37] is called Projected LM (PLM). The basic version of
the LM algorithm has been suitably modified to handle the non-uniqueness
of a state-space representation that may cause ill-conditioning of matrices
during model estimation. In addition, parameter initialization is carried out
by means of a deterministic procedure, thus avoiding the dependence of the
estimated model to the initial guess of parameters. Roughly speaking, the
gradient direction search in the PLM algorithm is modified to avoid the di-
rections in the parameter space that do not change the cost function due to
a similarity transformation of model matrices (see details in [35, 37]). The
initial guess of the parameters defining the local models are set equal to
the matrices of a single global stable linear model. The parameters of the
global linear model are computed by means of the application of an efficient
subspace identification method of the 4SID class [47]. The latter subspace
method also provides the automatic computation of the number n of internal
state variables, i.e., the size of vector z in (4.7) and the initial condition of
the state variables. Besides, the initial radial weighting functions ρj (·) are
distributed uniformly over the range of the input sequence. Using linear mod-
els for initialization is a nice feature that has been proposed and motivated
in [48]. It is worth to remark that the initialization by means of 4SID method
provides a bounded-input, bounded-output stable initial model. As pointed
out in [35, 37], the proposed PLM algorithm likely leads to the identification
of models that are locally linear stable. This is due to the choice of the
normalized weights along with the form of the scheduling vector accounting
for present and past samples of the input sequence only. Nevertheless, no
additional constraints are included to enforce stable models during training
and local stability can only be verified a posteriori.
28
35. 4 – Model representations
The number of local linear models p and the number of input past samples
accounted by the scheduling vector r must be selected before the estimation.
Again, we can choose those values by comparing models with increasing p
and r values, with respect to the estimation accuracy. Obviously, good p
and r values are the smallest ones leading to a good reproduction of the
identification signals.
4.6 Summary
In this section, we briefly compare the general strengths and limitations
of the model representation described throughout this Chapter. Table 4.1
summarizes the performances related to the estimation time, model size and
stability for the three different models under consideration. An up-arrow
“↑” means a positive feature, conversely a down-arrow “↓” means a negative
feature.
estimation model local
model
time size stability
SBF ↓ ↑↑ ↓ (a-posteriori)
ESN ↑ ↓ ↑ (a-priori)
LLSS ↓ ↑ ↑ (a-posteriori)
Table 4.1. Comparison of the features of the model representations con-
sidered.
From the figures in Table 4.1 it is clear that we cannot choose a-priori the
best candidate for model representation. A systematic study of the model
performances for IC ports must be carried out. This is done in Chapter 5.
29
36. Chapter 5
Assessment of models
5.1 Introduction
In this section, we assess the performances of the model representations de-
scribed in Chapter 4 by applying the modeling process proposed in Chapter 3
to the port voltage-current relation of a synthetic nonlinear dynamic one-port
test device relation. The reference model is the ODE (Ordinary Differential
Equations) description of the test device driven by a real voltage source. The
reference model is used to compute the responses needed for the model iden-
tification and for model validation. At the end of the analysis, we select the
model representation that is most feasible for the modeling of IC ports.
5.2 Test device
This section describes and discusses the structure of synthetic nonlinear dy-
namic one-port test device being modeled. Figure 5.1 shows the circuit
schematic of the test device, where the external port voltage and current
are v and i respectively. This device is designed in order to be represen-
tative of the IC input/output buffers such as drivers and receivers. It is
composed by a cascade connection between the two-port element represent-
ing a realistic package of an IC and the one-port element representing the
nonlinear functional part of the active device. The common port connecting
the package and the nonlinear functional part is identified by the voltage v1
and current i1 . The package is modeled by a lumped network of elements
30
37. 5 – Assessment of models
Rpkg , Lpkg and Cpkg , whose values are those tabulated for the standard pack-
age TSSOP48. The nonlinear functional part is mainly characterized by the
voltage-controlled current source f1 (v1 ), which defines the type of device op-
eration. When f1 = 0, the test device addresses a receiver input port. On
the contrary, when f1 = 0 the test device represents the output port of a
driver in a fixed logic state with f1 as the port static characteristic. In either
receiver or driver configuration, the voltage-controlled current source f2 (v2 )
attempts to mimic the typical mechanism of ESD (ElectroStatic Discharge)
protection of silicon devices. In particular, f2 is described by a nonlinear sta-
tic characteristic that works decreasing the output current i1 whenever the
port voltage v1 is larger than a threshold set by the battery VDD . Finally,
the capacitor C represents the equivalent port capacitance of the functional
part and R and L account for the bonding wire link between VDD and f2 .
v2 (t)
i(t) i1 (t) L
Rpkg Lpkg f2 (v2 )
R
v(t) Cpkg v1 (t) C f1 (v1 )
VDD
Package Functional part
Figure 5.1. Schematic of the one-port test device under modeling.
For the design of the device in Figure 5.1, our goal is to set the circuit
elements in a way to obtain a stiff modeling benchmark. In particular, we
carefully design the nonlinearity of f2 and we choose the value of L, since such
parameters mainly influence the nonlinear dynamic behavior of the device.
We consider the case of driver operation in a fixed logic state with a voltage
threshold for the activity of f2 fixed by VDD = 1 V. The characteristic of
the voltage-controlled current sources f1 and f2 are chosen in order to be
representative for the typical nonlinearities of silicon devices. In particular,
for the characteristic of f1 we select the parametric equation
f1 (v1 ) = a1 − a2 e−a3 v1 − a4 v1 (5.1)
31
38. 5 – Assessment of models
whereas for the characteristic of f2 we select the parametric equation
f2 (v2 ) = b1 eb2 (v2 −VDD ) (5.2)
The tuning of parameters a1 , a2 , a3 and a4 in equation (5.1) is done in
order to fit the typical current range of the most common technologies of
digital IC drivers, with respect to the chosen value of VDD . Conversely, the
parameters b1 and b2 of equation (5.2) are devised to observe a pronounced
activity of f2 whenever the voltage v2 is larger than zero. The final shape of
the characteristics (5.1) and (5.2) are reported in Figure 5.2.
80
60
40
f1 mA
20
0
−20
−40
−0.5 0 0.5 1 1.5
v1 v
50
40
f2 mA
30
20
10
−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25
v2 v
Figure 5.2. Characteristics of the voltage-controlled current sources of the
one-port test device in Fig. 5.1: f1 (v1 ) described by the equation (5.1) (top
panel) and f2 (v2 ) described by the equation (5.2) (bottom panel).
The tuning of the value of inductance L is performed in order to observe
a different dynamic behavior between the situation in which f2 provides a
significant amount of current and the situation in which f2 provides a neg-
ligible amount of current. Table 5.1 groups the values of the parameters in
equations (5.1) and (5.2) and of the passive components in the circuit of
Figure 5.1.
32
39. 5 – Assessment of models
Element Value
a1 70 mA
a2 10 mA
a3 1.35 V−1
a4 30 mΩ−1
b1 550 A
b2 12 V−1
Rpkg 0.5 Ω
Lpkg 3 nH
Cpkg 0.1 pF
R 0.1 Ω
C 5 pF
L 1.5 nH
Table 5.1. Values of the parameters in equations (5.1) and (5.2) and of
the passive components composing the test device of Figure 5.1.
5.3 Modeling setup
In this Section we briefly review the application of the modeling process de-
scribed in Chapter 3 to the test device in Figure 5.1, by means of parametric
models. In particular for each step, we highlight the main elements and
features impacting on the performances of the obtained model. The next
subsections address the following steps of the process
(1) Model selection
(2) Identification signals
(3) Model estimation
(4) Model validation
Step (5), related to the model implementation in a circuit simulator, is
skipped since it is not relevant for the application at hand.
5.3.1 Model selection
As outlined in Section 3.2, the first step concerns the selection of model for
the test device. Among all possible choices, we select three main classes (see
33
40. 5 – Assessment of models
Chapter 4 for details)
(i) Sigmoidal Basis Functions (SBF)
(ii) Echo State Networks (ESN)
(iii) Local Linear State-Space (LLSS)
For each representation, the model structure must be selected according to
the two different choices presented in Section 3.2 (fully nonlinear or splitted,
details in Appendix A).
5.3.2 Identification signals
In this step, the collection of the identification signals is addressed. This is
done by applying a real voltage source to the one-port test device in Fig. 5.1
and by recording the corresponding port voltage and current by means of
a transient simulation. Such an experiment is described by the test setup
in Figure 5.3 where the port of the test device is driven by an excitation
composed of a voltage source vs and a series-resistor Rs .
Rs i(t)
one-port
vs (t) v(t) test
device
Figure 5.3. Modeling setup for the one-port test device in Fig. 5.1.
For achieving the transient simulation, the continuous-time ODE descrip-
tion of the test setup in Figure 5.3 is selected as the reference model. Such
an ODE description is formulated in the canonical form
˙
M (x) x = h (x,u)
(5.3)
y = cT x + dT u
34
41. 5 – Assessment of models
where y is the output variable corresponding to the port current i, x =
[ v , i1 , v1 , v2 ]T is the vector collecting the state variables, and u = [ vs , VDD ]T
is the input vector accounting for the time-varying and fixed voltage sources.
The first row of the formulation (5.3) is a nonlinear differential equation
describing the state evolution. This equation involves the nonlinear state-
dependent so-called mass matrix
1 0 0 0
0 1 0 0
M (x) = 0 0 1 (5.4)
0
df2 (v2 )
0 0 0
dv2
and the nonlinear functional depending on the state and input vectors
1 v vs
− + i1 +
Cpkg Rs Rs
1
(−v − Rpkg i1 + v1 )
Lpkg
h (x,u) =
(5.5)
1
(f1 (v1 ) − f2 (v2 ) − i1 )
C
1
(v1 − v2 − VDD − Rf2 (v2 ))
L
The second row of the formulation (5.3) is a linear algebraic equation de-
scribing the output variable from the state and input vectors by means of
the vectors c = [ 1/Rs , 0 , 0 , 0 ]T and d = [ −1/Rs , 0 ]T . The identification
signals v and i are obtained by simulating the test setup of Figure 5.3 by
solving an initial value problem for the ODE formulation (5.3). The solution
is numerically calculated by means of Matlab ODE functions [49].
According to the guidelines outlined in Section 3.3, the driving voltage
source vs is a multilevel signal with superimposed small noise. Such a signal
must be devised in order to obtain identification signals rich of information
on the device behavior, i.e., the signal vs must excite the dynamic behavior
of the device for values within all possible operating voltages, even extend-
ing outside the power supply rails. For this test, we select a series-resistor
Rs = 50 Ω and a voltage source vs composed of thirty levels spanning the
35
42. 5 – Assessment of models
range of operating voltage [0 V , VDD + ∆], where ∆ = 0.5 V is the accepted
overvoltage. As outlined in Sec. 3.3, the design of a multilevel stimulus re-
quired by the identification of nonlinear dynamical systems is a matter or
repeated experiments. For the modeling of digital devices we performed a
systematic set of experiments which confirms that the quality of the esti-
mated models is weakly sensitive to the parameters defining the multilevel
signals [19]. As an example, a number of levels within the range between
five to some tens is sufficient for leading to accurate models reproducing the
original system response well. The superimposed noise is of gaussian type
with a selected standard-deviation of 0.1 mV. The flat parts of vs last 6 ns,
i.e., a sufficient duration to allow the port to reach steady state operation.
The duration of level transition is set to 200 ps, i.e., a typical value for the
switching time of modern high-speed devices. The sampling period used to
discretize the signals is Ts = 10 ps. Figure 5.4 shows the multilevel voltage
sources previously described and the corresponding identification signals as
result of the numerical transient simulation of the test setup of Figure 5.3.
1.5
vs V
1
0.5
0
0 5 10 15 20
1.5
v V
1
0.5
0 5 10 15 20
50
i mA
0
−50
0 5 10 15 20
t ns
Figure 5.4. Waveforms for the identification test computed for the setup of
Figure 5.3. Noisy multilevel voltage source vs (t) (top panel), identification
signals v(t) (middle panel) and i(t) (bottom panel) are shown.
36
43. 5 – Assessment of models
5.3.3 Model estimation
In this step, the unknown parameters defining the model are estimated from
the identification signals collected in the previous step. As pointed out in
Section 3.4, the parameters are evaluated by minimizing a suitable error
function by means of a specific algorithm based on the type of mathematical
representation describing the model. In order to evaluate how an estimated
model is good in the reproduction of the identification signals, we compute
the Mean Squared Error (MSE), defined as
N
1
MSE = (¯(k) − y(k))2
y (5.6)
N
k=1
where y (k) is the sample of the reference response (in this case the port cur-
¯
rent of the test device) and y(k) is the sample of the model response. When
dealing with SBF and LLSS models, their estimation relies on the application
of LM-based iterative algorithms. In this case, the estimation ends when an
iteration decreases the MSE of an amount below a small threshold, a priori
defined (e.g., 1e-8).
5.3.4 Model validation
According to Section 3.5, accuracy, local stability and efficiency of the es-
timated model are addressed at this point. The accuracy of the models is
quantified by computing the MSE between the model and the test device,
for a validation test. The validation test consists in the transient simulation
of the test setup of Figure 5.3 based on the ODE formulation (5.3), where vs
is again a multilevel voltage source but different from the one used for the
collection of the identification signals and shown in Figure 5.4. In this case,
the voltage source vs is composed of ten levels in the same range spanned
by the voltage source depicted in Figure 5.4. The standard-deviation of the
superimposed noise and the duration of flat parts are kept to 0.1 mV and
6 ns respectively, whereas the duration of level transition is set to 100 ps.
The sampling period used to discretize the signals is Ts = 10 ps. Figure 5.5
shows the multilevel voltage sources and the corresponding validation signals
computed for the test setup of Figure 5.3.
As discussed in Section 3.5, the local stability is assessed by computing
37
44. 5 – Assessment of models
1.5
vs V
1
0.5
0
0 2 4 6 8 10
1.5
v V
1
0.5
0 2 4 6 8 10
50
i mA
0
−50
0 2 4 6 8 10
t ns
Figure 5.5. Waveforms for the validation test computed for the setup of
Figure 5.3. Noisy multilevel voltage source vs (t) (top panel), validation
signals v(t) (middle panel) and i(t) (bottom panel) are shown.
the eigenvalues of the respective linearized model equations [25] (see Ap-
pendix B). The eigenvalues are computed for each point explored by the
voltage and current responses of the models recorded during the transient
simulations of the validation test. With respect to a specific instant of time,
some of the eigenvalues may lie outside the unitary circle. The percentage of
eigenvalues that lie inside the unitary circle can be used as an indicator of
the model stability, for the test at hand. Finally, the efficiency is addressed
by quantifying the simulation time required by the different models, for the
validation test.
5.4 Model performances
This section describes the effects of the modeling process previously described
on the estimation and validation performances of the obtained models. In
particular, we investigate the influence of the model representation and of its
estimation algorithm.
38
45. 5 – Assessment of models
5.4.1 SBF model
Several tests carried out on the test device under modeling have demonstrated
that the performance of SBF models are the same for both fully nonlinear
and splitted structures. Therefore, for the sake of simplicity, the following
discussion focuses only on the fully nonlinear structure. For this application,
the dynamic order of the SBF model (see equations (4.2) and (4.5)) is set to
r = 3, since we verified that larger values do not lead to better models.
MSE of MSE of CPU time Local
run # p estimation validation for model stability
phase phase estimation s index %
1 6 2.02e-8 3.32e-7 22.0 99.7
2 10 5.46e-9 1.97e-6 15.3 99.8
3 6 5.85e-9 1.12e-5 1.7 96.8
4 7 2.65e-9 3.01e-6 3.0 99.1
5 6 1.81e-8 9.90e-7 1.4 98.3
6 9 2.93e-8 6.19e-7 15.1 99.9
7 7 2.45e-8 1.91e-6 20.2 99.1
8 7 2.17e-9 9.50e-7 3.6 99.7
9 8 5.96e-9 1.22e-6 5.0 99.5
10 9 3.58e-9 9.36e-7 2.8 98.4
Table 5.2. SBF model performances: model size, estimation and valida-
tion accuracy, model estimation time and local stability for ten different
runs by means of the SLM algorithm [42].
Table 5.2 collects the main figures related to the performances of SBF
models estimated by means of ten runs of the static version of the LM al-
gorithm (SLM) [42]. Each row is referred to a specific run and defines the
selected model size (p, column 2), the MSE achieved in estimation and vali-
dation phases (column 3 and 4), the CPU time required for the estimation of
the model (column 5) and the local stability index defined as the percentage
of the explored eigenvalues that are inside the unitary circle in the complex
plane (column 6). The figures in Table 5.2 highlight that the random initial-
ization of the SLM algorithm does not affect the estimation accuracy of SBF
models, since all the run provide comparable values of MSE in the estimation
phase. In particular, the MSE level achieved is proven to be satisfactory, as
39
46. 5 – Assessment of models
50
i mA
0
−50
0 5 10 15 20
t ns
Figure 5.6. Identification port current response. Reference (solid line),
and SBF model (dashed line) estimated by the SLM algorithm with the
run # 8.
shown in Figure 5.6, that compares the identification port current with the
response of the model with the best MSE value (run # 8 in Table 5.2). The
responses of the other models are not reported since they are very similar
to the best model response. On the contrary, the column of validation MSE
indicates how the pseudo-random initialization affects the accuracy of the
models when they are excited by a signal different from the one used for the
estimation. This remark is confirmed by the validation curves of Figure 5.7,
that compares the reference output port current response and the response
of all the SBF models listed in Table 5.2. From this comparison it is worth
noting that different models provide a quite large variability of response, but
the best model (run # 1 in Table 5.2) provides very good results. The last
column of Table 5.2, that collects the percentage of model eigenvalues outside
the unitary circle, highlights that all the estimated models exhibit a poten-
tial local instability during transient simulation. This is also confirmed in
Figure 5.8, showing the unitary circle in the complex plane and the position
of the eigenvalues explored during the validation test by the best model (run
#1). Besides, it is also clear that all the estimated models are very compact,
i.e., they are composed by a very limited number of basis functions in the
40
47. 5 – Assessment of models
range 6 ÷ 10 (see the second column of Table 5.2).
MSE of MSE of CPU time Local
run # p estimation validation for model stability
phase phase estimation s index %
1 10 2.07e-8 1.70e-4 88.4 12.9
2 10 1.77e-8 2.86e-5 181.5 96.5
3 7 2.22e-8 1.26e-6 157.6 99.8
4 10 8.56e-8 3.47e-6 91.1 37.7
5 4 2.00e-8 3.58e-6 59.1 98.8
6 6 2.81e-8 1.55e-6 83.2 99.7
7 10 1.56e-8 1.62e-6 186.8 99.5
8 9 1.80e-8 1.49e-6 175.0 98.6
9 7 1.90e-8 7.49e-7 67.2 99.4
10 5 5.22e-8 1.02e-6 70.3 76.6
Table 5.3. SBF model performances: model size, estimation and valida-
tion accuracy, model estimation time and local stability for ten different
runs by means of the RLM algorithm [43].
Similarly, Table 5.3 collects the same comparison for ten runs of the
application of the recurrent version of the LM algorithm (RLM) [43]. Similar
comments hold for the MSE during estimation, the model size and the local
stability. On the contrary, the estimated models exhibit a larger variability,
depending on the initial guess of parameters as shown in Figure 5.9. Finally,
from the comparison of the two estimation algorithm (SLM and RLM) it is
clear that the RLM method is less efficient and requires a larger CPU time
for the computation of the model parameters than the static version of the
method (compare the 5th column of Tables 5.2 and 5.3).
5.4.2 ESN model
Unlike the SBF case, the modeling with ESN representation has been proven
to be accurate only by using the splitted structure. Therefore, the fully
nonlinear structure is not considered hereafter. The ESN representation used
for modeling the problem at hand has been proven to be accurate enough by
selecting the state vector size n = 100 (see equation (4.6)).
41
48. 5 – Assessment of models
60
40
20
i mA
0
−20
−40
−60
0 2 4 6 8 10
t ns
Figure 5.7. SBF model validation: port current response of ten different
models obtained by means of the SLM estimation algorithm. Solid line:
reference, dashed line: best model (#1), dotted lines: other models.
MSE of MSE of CPU time Local
run # n estimation validation for model stability
phase phase estimation s index %
1 100 5.51e-6 9.19e-6 1.4 100
Table 5.4. ESN model performances: model size, estimation and valida-
tion accuracy, model estimation time and local stability for one run by
means of the estimation algorithm [36].
Table 5.4 collects the main figures related to the performances of the ESN
model estimated by means of the algorithm [36]. The performance indexes are
the same as those for the SBF case. However, we report the result of a unique
estimation run since we verified that the suggested initialization procedure
does not affect the quality of the obtained model. The local stability of the
model enforced a-priori is verified. As a confirmation, Figure 5.12 shows the
eigenvalues of the linearized ESN model for each point explored during the
transient validation test. From Table 5.4 we see that the value of the CPU
time required for the estimation is very small, since the estimation relies on
a linear least squares algorithm. Finally, Figure 5.11 shows the validation
42
49. 5 – Assessment of models
1
imaginary 0.5
0
−0.5
−1
−1 −0.5 0 0.5 1
real
Figure 5.8. Eigenvalues of the linearized SBF model providing the best
prediction accuracy shown in Fig. 5.7. The linearization is computed for
each point explored during the transient validation test.
curves, thus highlighting a good accuracy of the estimated ESN model. This
is also confirmed by the MSE value in Table 5.4.
5.4.3 LLSS model
The modeling of the test device by means of LLSS representation has been
proven to be accurate by using either the fully nonlinear or splitted struc-
ture. Therefore, as reported for the SBF case, for conciseness we discuss
only the performance of the fully nonlinear structures. For this application,
the scheduling vector of the LLSS model is composed by using r = 0 (see
equations (4.7) and (4.8)), since we verified that larger values do not lead to
better models.
Table 5.5 collects the main figures related to the performances of the
LLSS model estimated with one run by means of the PLM algorithm [37].
The performance indexes are the same as those for the SBF and ESN case.
We perform only one run of the estimation since the initialization with the
4SID method [47] is unique and deterministic. In addition, the initialization
also provides the dimension of the state vector of each local model, that is
n = 4 for this case. The reached MSE in estimation has been proven to be
43
